= Integral cohomology of a complex Stiefel manifold
{title2=$H^*(V_k(\mathbb C^n);\mathbb Z)=\Lambda(a_{2n-1},\ldots,a_{2n-2k+1})$}
Use the <complement bundle on a complex Stiefel manifold> inductively. If the ring for $V_k$ is exterior with smallest positive degree $2n-2k+1$, the complement bundle has Euler degree $2(n-k)$, in which base cohomology vanishes. Its Euler class is consequently zero. The <Gysin ring splitting for an odd-dimensional sphere bundle> adds one exterior generator of degree $2(n-k)-1$. The initial base $V_0$ is a point. This gives the full integral <cohomology ring> and in particular its torsion-freeness. At $k=n$, the <unitary group> has Poincare polynomial $\prod_{j=0}^{n-1}(1+t^{2j+1})$.
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